A couple years of years ago I calculated population-weighted density for different geographies in the US based on 2010 census tract data. My calculation had the national figure at 5362 people/sq mi, close to the official Census estimate of 5369. Here are some other insights:

First, a note on why population-weighted density is important. Regular population density figures divide the number of people by the size of the land, regardless of how the people are dispersed upon it. In a place like America, which has lots of empty land, this distorts things.

As an illustration, this map by cartographer Nik Freeman shows census tracts in dark green that have 0 population, a land area totaling 47% of the United States. Counting all this empty land puts our raw population density at a mere 87/sq mi! 5369 v. 87 is a massive difference.

Population-weighted density doesn't count all of this empty land, because tracts are weighted by how much of the larger total (counties, metro areas, states, nations) they comprise. This gives us a much better understanding of the level of density an average person experiences!

So as I said, my calculations put the population-weighted density of the United States at 5362 ppsm (people per square mile). But because this is an average, most people don't actually live at that level of density.

If you sort census tracts by density and do a rolling population sum, you find that only 24.1% of Americans live in census tracts with densities of 5362 ppsm or higher. The 50th percentile person lives in a census tract with the density of 2120 ppsm!

For comparison, here are smaller geographies that align with those national pop-weighted density figures:
5362- Dallas County, TX; Richmond, VA; the Baltimore metro area; the state of Nevada
2120- Polk County, IA (Des Moines); the Atlanta metro area; the state of New Mexico

Going into pop-weighted density figures for smaller geographies, we can start at the state-equivalent level: Even with the relatively rural living pattern Upstate, New York is easily the densest state in the nation, while Mississippi is the sparsest. NY is 40.5x as dense as MS.

New York's lead by this metric holds at the metro area level as well. Ranking the top 50 most populous metro areas by population-weighted density, NYC stands alone at 2.5x the density of San Francisco and Los Angeles (which, perhaps surprisingly to most, are essentially equal).

You can see that automobile-oriented Southern and Midwestern metro areas are the least dense, while 9 of the top 12 densest metros have rapid transit systems (NYC, SF, LA, Chicago, Boston, Philly, Miami, DC, Baltimore). [San Juan was too difficult to calculate at the metro level]

At the county-equivalent level by population-weighted density, Manhattan is easily tops, 50% denser than The Bronx in the 2nd position and 3.3 million (!) times denser the sparsest place in the country, the Yukon-Koyukuk Census Area in Alaska, our largest county by land area.

(seriously, the Yukon-Koyukuk Census Area is 147,805 square miles, the size of Montana. It would be the 4th biggest state by its own right! It only has 5,327 people. My census tract in Brooklyn has 4,630 people on 0.065 square miles...)

If you rank every county by population-weighted density, the state with the largest gap between its densest and sparsest county is Nevada. Clark County (Las Vegas) is 44th, while Esmeralda County, a mountainous area near Death Valley, is 3216th, a gap of 3172.

The state with the smallest ranking gap between its densest and sparsest county is Rhode Island. Providence County is 37th, while Washington County (South Kingstown) is 814th, a gap of only 777. Washington County is the highest-ranked low county of any state.

The state with the lowest-ranked high county is Missisippi, which as I stated earlier is the sparsest state in the country. Hinds County, home to the state capital and biggest settlement, Jackson, is 402nd at 1929.89 ppsm.

By my calculations, the nation's population-weighted density declined by 4.59% between the 2000 and 2010 census. 24 state-equivalents decreased in population-weighted density over that time, while 28 increased.

The state that had the largest increase in population-weighted density between 2000 and 2010 was Virginia, which densified by 17.7%, while the worst loss of density occurred in Louisiana (no doubt because of various climatic disasters), which declined by 23.1%.

Virginia's raw population density increased by 13% over that timeframe, meaning that population-weighted density increases outpaced raw population growth. 7 other states experienced this phenomenon. Vermont's factor was the largest, growing 8.3% pop-weighted and 2.8% raw.

The other states where population-weighted density gains outpaced raw population increases between 2000 and 2010 were South Dakota, Massachusetts, Montana, New Mexico, Oregon, and Arkansas.

I don't really have a conclusion here, so I guess I'll just say, yay geography, yay [population-weighted] density!

Another interesting thing is how tracts are distributed within metro areas. If you calculate various percentiles of tract density within metros and rank them against each other, you can see the "shapes" of cities. The 40th percentile tract in the LA Metro is denser than in NYC!

You can see Boston is extremely dense in its dense parts, but 20th + 40th percentile are middling. Similar pattern for Baltimore, Milwaukee. Contrast with Vegas, San Diego, etc. which fall in rankings as you reach the cores. Western cities have sparser cores but denser sprawl!

For the record I think it is much, much better to have extremely dense cores and sparsely populated hinterlands than to have a consistent level of dense sprawl across 10 counties in a metro area. All people should live in pedestrian-friendly hyper dense urban areas or on farms

The Baltimore metro has its fair share of atrocious low-to-mid density auto-suburbia (like basically all of Howard County), but it is very much a very dense port city with a lot of fairly dense streetcar suburbs surrounded by farmland (northern BalCo, Carroll, Harford Counties)

For context the 20th percentile density census tract in the Los Angeles metro area is 4,944.6 ppsm and 80th percentile is 17,742.2, for NYC those figures are 2,626.5 and 48,499.3. NYC is less dense in the hinterlands and far more dense at the core

San Francisco and Los Angeles metros were very close in my pop-weighted density calculations, but they are not shaped exactly the same. San Francisco overtakes Los Angeles in terms of density at their respective 77th percentile tracts and doesn't give the lead up.

This is how census tracts are ordered by density in the Northeast. Other 4 are very comparable until about the 57th percentile, then Boston pulls ahead. Philly starts to break away from Bmore+DC @~67th percentile. DC gets separation from Bmore at 88th, but Bmore's densest > DC's.